Assume that hypothesis \((\dagger _{n,\Lambda })\)  and \((\dagger \dagger _{n,\Lambda })\)  hold for \(\Lambda \in (\lambda (\mathbb {S}^{n}),2]\)  fixed. Consider a smooth closed hypersurface \(M^{n}\subset \mathbb {R}^{n+1}\)  with \(\lambda (M) \le \Lambda \) . Flowing \(M\)  by mean curvature flow for a short time strictly decreases the entropy unless \(M\)  is homothetic to a self-shrinker. If \(M\)  is homothetic to a self-shrinker other than \(\mathbb {S}^{n}(\sqrt{2n})\)  then by [1]}, a small \(C^{\infty }\) -perturbation of \(M\)  has strictly smaller entropy.
